JHEP01(2015)114

Published for SISSA by Springer

Received: October 6, 2014

Accepted: December 19, 2014

Published: January 21, 2015

Spherical Hecke algebra in the Nekrasov-Shatashvili

limit

Jean-Emile Bourgine

Asia Pacific Center for Theoretical Physics (APCTP),

Pohang, Gyeongbuk 790-784, Republic of Korea

E-mail: jebourgine@apctp.org

Abstract: The Spherical Hecke central (SHc) algebra has been shown to act on the

Nekrasov instanton partition functions of N = 2 gauge theories. Its presence accounts

for both integrability and AGT correspondence. On the other hand, a specific limit of the

Omega background, introduced by Nekrasov and Shatashvili (NS), leads to the appearance

of TBA and Bethe like equations. To unify these two points of view, we study the NS limit

of the SHc algebra. We provide an expression of the instanton partition function in terms

of Bethe roots, and define a set of operators that generates infinitesimal variations of the

roots. These operators obey the commutation relations defining the SHc algebra at first

order in the equivariant parameter ε2. Furthermore, their action on the bifundamental

contributions reproduces the Kanno-Matsuo-Zhang transformation. We also discuss the

connections with the Mayer cluster expansion approach that leads to TBA-like equations.

Keywords: Supersymmetric gauge theory, Duality in Gauge Field Theories, Quantum

Groups, Bethe Ansatz

ArXiv ePrint: 1407.8341

Open Access, c© The Authors.

Article funded by SCOAP3.doi:10.1007/JHEP01(2015)114

JHEP01(2015)114

Contents

1 Introduction 2

2 Spherical Hecke central algebra 3

2.1 Rewriting the SHc algebra 3

2.1.1 Partition function 3

2.1.2 Instanton positions and Λ-factors 4

2.1.3 Generators 6

2.2 KMZ transformation 8

3 SHc in the NS limit 9

3.1 Derivation of the Bethe equations 9

3.2 NS limit of the algebra 12

3.3 Covariance of the bifundamental contribution under SHc transformations 13

4 Connections with the Mayer cluster expansion 15

4.1 Integral expression of the A2 partition function 15

4.2 NLIE, Bethe roots and Yang-Yang functional 16

4.3 Hadronic integrals 19

4.3.1 Hadronic variables and Bethe roots 20

4.3.2 Relation between bifundamental summands and integrands 22

5 Discussion 23

A Computing with instanton positions 24

A.1 Transition of notations 24

A.2 Useful formulas 25

B Justifications 26

B.1 Order of Λx(Y ) 26

B.2 NS limit of the bifundamental contribution 26

C Nekrasov-Shatashvili limit from Mayer expansion 28

C.1 Equations of motion, grand-canonical free energy 28

C.2 Confinement 31

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JHEP01(2015)114

1 Introduction

In many respects, degenerate affine Hecke algebras seem to be the key behind the fascinat-

ing structure of instanton partition functions of 4d N = 2 gauge theories. The Spherical

Hecke central (SHc) algebra constructed in [1] is a limit of a symmetrized double degenerate

Hecke algebra (DDAHA). It acts on instanton partition functions as the Kanno-Matsuo-

Zhang (KMZ) transformation of the Young diagrams summand, also called bifundamental

contribution [2]. SHc representations contain WN subalgebras which is the main reason

behind the AGT correspondence [3, 4]. And indeed, the various proofs of AGT corre-

spondence exploit this underlying algebraic structure. For instance, a set of (generalized)

Jack polynomials have been used in the proofs [5–9] based on the free field representation

of CFT conformal blocks [10, 11]. Jack polynomials are known to be the eigenstates of

the Calogero-Moser Hamiltonian which is one of the SHc generators in the polynomial

representation relevant to the formal case of gauge groups SU(Nc) with Nc = 1 [1]. The

SHc algebra has a Hopf algebra structure, and it is possible to employ the comultiplica-

tion to define a representation of the generators relevant to the case of a higher number

of colors (Nc > 1). This leads to define generalized Jack polynomials which diagonalize

the Calogero-Moser Hamiltonian acting in a tensor space [8, 12]. The first proof of AGT

correspondence is also based indirectly on SHc. It uses a specific basis of the CFT Fock

space formed by the AFLT states. These states were defined such that the decomposition

of conformal blocks coincides with the expression of Nekrasov partition functions [13–16].

They were later identified with the generalized Jack polynomials [12, 17] using the free

field construction presented in [18, 19].

DDAHA, from which SHc is constructed, is formed by Dunkl operators, and contains

an infinite number of commuting integrals of motion. It has a deep connection with the

quantum inverse scattering method [20–22], which may lead to a better understanding of

the instanton R-matrix proposed by Smirnov [23] and built upon the stable map intro-

duced in [24].

In this paper, we are interested in yet another aspect of SHc. In [25], Nekrasov and

Shatashvili (NS) have investigated a limit of the Ω-background in which one of the equiv-

ariant parameters tends to zero. They have proposed to express the gauge theory free

energy in this limit using the solution of a non-linear integral equation (NLIE) reminiscent

of those derived from the Thermodynamical Bethe Ansatz [26, 27]. This proposal was con-

firmed in [28, 29], exploiting the Mayer cluster expansion technique to perform the subtle

limit. In a different study of the NS limit, Bethe-like equations and TQ-relations were

also obtained [30–33]. In both cases, the associated integrable model bare some similarities

with the sl(N) XXX spin chain, and the system of bosons with delta interaction studied

by C. N. Yang [34].1 The latter also describes the solution of the quantum non-linear

Schrodinger equation in a sector of fixed number of particles, and will be referred shortly

as qNLS. Both integrable models are related to Hecke algebras, although in a different way.

1Both systems share many common properties. For instance, Yang-Yang functionals have the same

quadratic part, and differ only by the potential term. Coordinate wave functions are also very close. The

gauge theory seems to appeal to what these two systems have in common.

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JHEP01(2015)114

XXX spin chains (with open boundaries) are known to possess a Yangian symmetry, which

is a representation of the degenerate affine Hecke algebra [35]. On the other hand, the

qNLS Hamiltonian can be diagonalized by a technique involving Dunkl operators. In the

case of periodic boundary conditions, these operators form with the affine Weyl group a

DDAHA [36]. In view of these results, it seems necessary to understand the fate of the

SHc algebra in the NS limit, and possibly relate it to the algebraic structure of XXX and

qNLS integrable models. This paper reports on the first step in this direction.

We focus on a A2 quiver gauge theory with U(Nc)×U(Nc) gauge group, Nc fundamental

flavors at each node, and a bifundamental matter field. This theory is asymptotically

conformal. We regard more precisely the bifundamental contribution to the Nekrasov

instanton partition function. This quantity depends on two sets of Nc Young diagrams,

and is a building block for the instanton partition function of more general quivers. The

KMZ transformation represents the action of SHc generators as a variation of the number

of boxes in the Young diagrams. In order to proceed to the NS limit, the action of these

generators must be re-written in a suitable manner. This is done in the second section. In

the third section, we recall the derivation of the Bethe-like equations from the invariance

of the Young diagrams summands under a variation of boxes. It leads to identify the Bethe

roots with the instanton positions of the boxes on top of each column. This identification

further provides an expression of the bifundamental contribution. We then construct a set

of operators upon infinitesimal variations of the Bethe roots, and show that they obey the

SHc commutation relations at first order in ε2. We also recover the KMZ transformation

in this limit. Finally, in a fourth section, we explain the connection with the outcome

of the Mayer cluster expansion, and TBA-like NLIE. Technical details are gathered in

the appendices.

2 Spherical Hecke central algebra

The main properties of the SHc algebra can be found in [2] which is the starting point for

our rewriting process. In order to render the ε2 factors explicit, we will avoid the notations

β and ξKMZ = 1− β, and use instead directly the Ω-background equivariant parameters ε1and ε2, together with the shortcut notation ε+ = ε1 + ε2. These parameters are related to

the previous quantities through β = −ε1/ε2 and ξKMZ = ε+/ε2. Let us emphasize that no

limit is taken in this section, and expressions are exact in ε1 and ε2.

2.1 Rewriting the SHc algebra

2.1.1 Partition function

The building block of quiver partition functions is the so-called bifundamental contribu-

tion.2 It is associated to an arrow of the quiver between two nodes a → b. It depends on

the information at each node a, encoded in an object Ya, and a mass mab of bifundamen-

tal matter fields. Here we focus on the A2 quiver, with two nodes a = 1, 2, and a single

2Strictly speaking, this quantity also contains a contribution from the vector multiplets coupled to the

bifundamental multiplet in its denominator. These normalization factors simplify the expression of the

KMZ transformation.

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JHEP01(2015)114

fundamental mass m for which we drop the index 12. To each node a corresponds a gauge

group U(N(a)c ), and a vector of Coulomb branch vevs a

(a)l with l = 1 · · ·N (a)

c the color

index. We further associate to a node a the object denoted Ya that consists in a set of N(a)c

Young diagrams Y(l)a , together with the vector a

(a)l . Young diagrams Y

(l)a are made of n

(a)l

columns λ(a,l)i with λ

(a,l)i+1 ≤ λ

(a,l)i . The dual partition consists of the sequence of integers

λ(a,l)i = ]ji ≤ λ

(a,l)j for i = 1 · · ·λ(a,l)

1 . Indices i of λ(a,l)i and λ

(a,l)i can be extended to

infinity, setting the additional quantities to be zero. For simplicity, we assume that both

nodes have the same gauge group, i.e. N(1)c = N

(2)c = Nc.

The bifundamental contribution can be expressed in terms of the variables t(a)l,i and the

dual ones t(a)l,i associated to Ya, and defined as

t(a)l,i = a

(a)l + ε1(i− 1) + ε2λ

(a,l)i , t

(a)l,j = a

(a)l + ε1λ

(a,l)j + ε2(j − 1), (2.1)

where i = 1 · · ·n(a)l spans the columns, and j = 1 · · ·λ(a,l)

0 the rows of the diagram Y(l)a .

It reads

Z[Y1, Y2] =

∏l,(i,j)∈Y1

∏Ncl′=1(t

(1)l,j − t

(2)l′,i − µ+ ε2)

∏l,(i,j)∈Y2

∏Ncl′=1(t

(2)l,j − t

(1)l′,i + µ− ε1)∏

a=1,2

[∏l,(i,j)∈Ya

∏Ncl′=1(t

(a)l,j − t

(a)l′,i + ε2)(t

(a)l,j − t

(a)l′,i − ε1)

]1/2,

(2.2)

with |Ya| the total number of boxes of Young diagrams. This expression has been obtained

from the equation (6) of [2] under a rescaling of Coulomb branch vevs and bifundamental

mass: a(a)l = −ε2ap,KMZ and µ = −ε2µKMZ.3 The variable µ corresponds to a shifted

bifundamental mass, it is related to m through µ = ε+/2−m. In the following we mostly

focus on a single object Ya=1 and, to alleviate the notations, the label a will be dropped

when no ambiguity arise.

2.1.2 Instanton positions and Λ-factors

The SHc algebra found in [2] is acting on states |Y > characterized by an object Y (we

dropped the node index a). To each box x ∈ Y is associated a triplet of indices l, i, j

such that (i, j) ∈ Y (l) is a box in the lth Young diagram. In our reformulation of the SHc

algebra, a prominent role is played by the instanton position which is a map that associates

to each box x ∈ Y the complex number

φx = al + (i− 1)ε1 + (j − 1)ε2. (2.3)

Thus, the variables tl,i defined in (2.1) correspond to the instanton positions of the box

locations that lay on top of each column λ(l)i . Similarly, tl,j is the instanton position of a

box located directly on the right of each row.

We denote A(Y ) (resp. R(Y )) the set of boxes that can be added to (resp. removed

from) Y . The set A(Y ) is a subset of the locations with positions tl,i, and also of the

3This rescaling is necessary in order to have a non-trivial dependence. Correspondingly, the study of

semiclassical Liouville correlators with heavy operators is more involved than the case of light ones.

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JHEP01(2015)114

locations with positions tl,j . Likewise, the set R(Y ) is a subset of the locations with

positions tl,i − ε2, and tl,j − ε1:

A(Y ) ⊂ x ∈ Yφx = tl,i, A(Y ) ⊂ x ∈ Yφx = tl,j,R(Y ) ⊂ x ∈ Yφx = tl,i − ε2, R(Y ) ⊂ x ∈ Yφx = tl,j − ε1.

(2.4)

Note that for each Young diagram the number of boxes one can add is always one plus

the number we can remove: ]A(Y ) = ]R(Y ) + Nc where Nc is the number of Young

diagrams in Y .

In the original paper [2], the KMZ transformation is expressed using a decomposition

of Young diagrams into elementary rectangles. This rectangle decomposition is useful to

characterize the coordinates of boxes in the sets A(Y ) and R(Y ). However, it turns out

that this information is not essential to the formulation of the KMZ transformation. It is

sufficient to use the instanton positions φx for x ∈ A(Y ) or x ∈ R(Y ) without specifying

the coordinates of these boxes in the Young diagrams. Omitting this information results in

simpler expressions for the SHc generators and the KMZ transformation. The transition

between the notations of the KMZ paper [2] and ours is explained in the appendix A. It

may be summarized by the formulas (A.1) and (A.2).

One of the main simplifications in our formalism concerns the factors Λ(k,±)l given by

equ (12) and (13) in [2] that can be merged into a single quantity,

Λx(Y )2 =∏

y∈A(Y )y 6=x

φx − φy + ε+φx − φy

∏y∈R(Y )y 6=x

φx − φy − ε+φx − φy

. (2.5)

Note that the term y = x has to be removed from the left product if x ∈ A(Y ) and the right

one for x ∈ R(Y ). The expression remains valid if x /∈ A(Y ) ∪ R(Y ) with both products

complete. In the limit ε+ = 0 (or β = 1), we trivially have Λx(Y ) = 1. The factors Λx(Y )

play a major role in the definition of the SHc algebra, and it is important to study their

properties. They correspond to the residues at the poles of the function

Λ(z)2 =∏

x∈A(Y )

z − φx + ε+z − φx

∏x∈R(Y )

z − φx − ε+z − φx

, (2.6)

which can be decomposed as

Λ(z)2 = 1 + ε+∑

x∈A(Y )

Λx(Y )2

z − φx− ε+

∑x∈R(Y )

Λx(Y )2

z − φx. (2.7)

An infinite tower of identities can be obtained through an expansion at z →∞. The first

identities are given in the appendix A.

Operators of the SHc algebra act on the states |Y > by adding or removing boxes.

We denote Y + x and Y − x the object Y with a box x ∈ A(Y ) (resp x ∈ R(Y )) added to

(removed from) the set of Young diagrams. The behavior of the coefficients Λx(Y ) under

such a procedure is remarkably simple:

Λy(Y − x)2 = r(φx − φy)Λy(Y )2, Λy(Y + x)2 = r(φy − φx)Λy(Y )2, (2.8)

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JHEP01(2015)114

where we introduced the function

r(z) =(z + ε1)(z + ε2)(z − ε+)

(z − ε1)(z − ε2)(z + ε+), r(−z) = 1/r(z). (2.9)

It is important to note that the quantity r(φx−φy) is independent of the Young diagrams.

Let x ∈ R(Y ), the first formula in (2.8) is valid if y ∈ A(Y − x) ∪A(Y ) or y ∈ R(Y − x) ∪R(Y ). It breaks down if y ∈ R(Y − x) but y /∈ R(Y ) which corresponds to the poles of

r(φx − φy) for φx − φy = ε1 or ε2 (boxes under x or on the left). Extending the definition

of Λy(Y )2 to x /∈ Y , r(z) in (2.8) has to be replaced by its residue divided by ε+. When

y ∈ A(Y − x) but y /∈ A(Y ), i.e. y = x, we have to replace r(0) = −1 by one. The same

comment is valid for x ∈ A(Y ): if y ∈ A(Y +x) but y /∈ A(Y ), it implies that φy −φx = ε1or ε2 (boxes above x or on its right) and the pole of r(z) has to be replaced by the residue

divided by ε+. For y = x ∈ R(Y + x), we have Λx(Y + x)2 = Λx(Y )2.

2.1.3 Generators

We are now ready to introduce the generators Dn,m of the SHc algebra. The index n runs

over all integers and is referred as the degree, whereas the second index m is a positive

integer called the order. The action of these generators on the states |Y > is known in a

closed form only for the operators of degrees 0 and ±1. For those operators, we introduce

the generating series

D±1(z) =∞∑n=0

(z + ε+)−n−1εn2D±1,n, D0(z) =∞∑n=0

(z + ε+)−n−1εn2D0,n+1. (2.10)

The construction of these operators from the degenerate affine Hecke algebra is done in [1].

For Nc = 1, the generators D0,n in the polynomial representation are symmetric polynomi-

als of the Cherednik(-Dunkl) operators for the Calogero-Moser model. In particular, D0,2

coincides with the Hamiltonian of this integrable system, and the states |Y > can be iden-

tified with the eigenbasis of Jack polynomials. The operators D±1,n are built by exploiting

the Pieiri formula satisfied by the Jack polynomials [37]. At Nc > 1, it is possible to exploit

the Hopf algebra structure of SHc to define a tensor representation for D0,n [1]. Then, the

states |Y > should be identified with the generalized Jack polynomials which diagonalize

the comultiplication of D0,2 [8, 12, 17].

The action of the generators on states |Y > is defined by the formulas (38)-(40) of [2],

which can be rewritten as

D±1(z)|Y >=∑

x∈A/R(Y )

Λx(Y )

z − φx|Y ± x >, D0(z)|Y >=

∑x∈Y

1

z − φx|Y > . (2.11)

The commutation relations (22)-(23) of [2] can be summed up to produce

[D0(z), D±1(w)] = ±D±1(w)−D±1(z)

z − w, [D−1(z), D1(w)] =

E(w)− E(z)

z − w, (2.12)

where we defined another generating series:

E(z) =∞∑n=0

(z + ε+)−n−1εn2En. (2.13)

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JHEP01(2015)114

The specific properties of the SHc algebra comes from the expressions of the generators

En as a function of degree zero operators. They are encoded in the identity

1 + ε+E(z) = exp

∑n≥0

(−1)n+1cnπn(ε2/(z + ε+))

exp

∑n≥0

D0,n+1ωn(ε2/(z + ε+))

,

(2.14)

with the functions

πn(s) = snGn(1 + sε+/ε2), ωn(s) =∑

q=ε2,ε1,−ε+

sn (Gn(1− qs/ε2)−Gn(1 + qs/ε2)) ,

G0(s) = − log(s), Gn(s) = (s−n − 1)/n, (n ≥ 1). (2.15)

The central charges cn are the Miwa transformed of the Coulomb branch vevs,

εn2cn = (−1)nNc∑l=1

(al + ε+)n. (2.16)

It is possible to deduce the action of E(z) on states |Y > from (2.14) and the action (2.11)

of D0(z):4

(1 + ε+E(z)) |Y >=

Nc∏l=1

(1 +

ε+z − al

) ∏x∈Y

r(z − φx)|Y >, (2.18)

with the function r(z) defined in (2.9). In the r.h.s. , the product over r(z − φx) and

Coulomb branch vevs reduces to the function Λ(z)2 as a result of the identity5

∏x∈Y

r(z − φx) = Λ(z)2Nc∏l=1

z − alz − al + ε+

. (2.19)

Operators of higher degree are constructed using the following commutation relations

(n ≥ 0,m > 0):

D±(n+1),0 = ± 1

n[D±1,1, D±n,0], D±n,m = ±[D0,m+1, D±n,0]. (2.20)

From these commutation relations, the generators of degree zero and ±1 engender the whole

algebra. In particular, the U(1) and Virasoro subalgebras are generated by the modes of

the current Jn and stress-energy tensor Ln defined with generators of order zero and one:

J±n =(−√−ε1/ε2

)−nD∓n,0, L±n =

1

n

(−√−ε1/ε2

)−nD∓n,1 +

1

2(1− n)Nc

ε+ε2J±n.

(2.21)

4We have used the formula

log

(1 + (a+ b)s

1 + as

)=∑n≥0

(−1)n+1ansnGn(1 + bs). (2.17)

5This identity is easy to derive for Young diagrams that are rectangles. It proceeds from the numerous

cancellations between numerators and denominators in the l.h.s. product. For an arbitrary Young diagram,

the equality is obtained by decomposition into elementary rectangles.

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JHEP01(2015)114

Figure 1. Location of the operators Dn,m in the Z × Z+ half-plane. In green, the generators

forming D0(z) and D±1(z) for which the action on the states |Y > is known. In blue and red,

the modes of the U(1) current and stress-energy tensor respectively. Black arrows represent the

commutation relations (2.20).

A summary of the different relations between the SHc generators can be found in figure 1.

The generators of Virasoro and U(1) subalgebras are orthogonal to the operators composing

the series D0(z) and D±1(z). The standard current and Virasoro modes commutation

relations can be recovered from (2.12) and (2.20). However, this nested calculation rapidly

becomes impractical.

2.2 KMZ transformation

The KMZ transformation is generated by the operators δ±1,n in [2]. Here, we use instead

the generating series

δ±1,z =

∞∑n=0

(z + ε+)−n−1(−ε2)nδ±1,n. (2.22)

In our notations, they act on the bifundamental contributions (2.2) as

δ−1,zZ[Y1, Y2] =∑

x∈A(Y1)

Λx(Y1)

z − φxZ[Y1 + x, Y2]−

∑x∈R(Y2)

Λx(Y2)

z − µ− φxZ[Y1, Y2 − x], (2.23)

δ+1,zZ[Y1, Y2] = −∑

x∈R(Y1)

Λx(Y1)

z − φxZ[Y1 − x, Y2] +

∑x∈A(Y2)

Λx(Y2)

z + ε+ − µ− φxZ[Y1, Y2 + x].

The operator δ−1,z combines an action of D+1(z) on the state |Y1 > (first term) and an

action of D−1(z−µ) on |Y2 > (second term). In δ+1,z, the actions of D±1(z) are exchanged:

D−1(z) now acts on |Y1 > and D+1(z + ε+ − µ) on |Y2 >. It is possible to work at fixed

box x using a contour integral circling φx. For x ∈ A/R(Y1), we have∮φx

dz

2iπδ∓1,zZ[Y1, Y2] = ±Λx(Y1)Z[Y1 ± x, Y2]. (2.24)

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JHEP01(2015)114

The covariance of the bifundamental contribution Z[Y1, Y2] under the KMZ transfor-

mation involves the functions

U−1,z =

∏x∈R(Y1)(z − ε+ − φx)

∏x∈A(Y2)(z − µ+ ε+ − φx)∏

x∈A(Y1)(z − φx)∏x∈R(Y2)(z − µ− φx)

− 1,

U+1,z =

∏x∈A(Y1)(z + ε+ − φx)

∏x∈R(Y2)(z − µ− φx)∏

x∈R(Y1)(z − φx)∏x∈A(Y2)(z − µ+ ε+ − φx)

− 1.

(2.25)

Note that the minus one in the r.h.s. subtracts the pole at z = ∞. Under the KMZ

transformation, Z[Y1, Y2] behaves as

√−ε1ε2δ±1,zZ[Y1, Y2] = U±1,zZ[Y1, Y2]. (2.26)

At fixed box x ∈ A(Y1) this relation writes

√−ε1ε2Λx(Y1)Z[Y1 + x, Y2]=

∏y∈R(Y1)(φx − φy − ε+)

∏y∈A(Y2)(φx−φy−µ+ ε+)∏

y∈A(Y1)y 6=x

(φx − φy)∏y∈R(Y2)(φx − φy − µ)

Z[Y1, Y2].

(2.27)

A similar relation can be obtained for x ∈ A(Y2) by exchanging Y1 ↔ Y2, and replacing

µ→ ε+ − µ.

3 SHc in the NS limit

We now recall the derivation of the Bethe equations in the NS limit along the lines of [33,

38]. The derivation exploits the covariance of summands of the partition function under

variation of the number of boxes in Young diagrams. We argue that the main contribution

to the partition function comes from Young diagrams with columns of infinite height, and

multiplicity one. The profile of these diagrams is fully encoded in a set of Bethe roots

which will be used to express the partition function.

3.1 Derivation of the Bethe equations

In [39], Nekrasov and Okounkov have exposed a procedure to compute the instanton parti-

tion function in the Seiberg-Witten limit ε1, ε2 → 0. In this limit, Young diagrams become

infinitely large, and are described by a continuous profile minimizing an effective action.

This method was later extended to the NS limit ε2 → 0 in [30–32, 40].6 It relies on the

idea that Young diagrams summations are dominated by a set of Young diagrams with a

specific profile. This profile is an extremum, which implies that small deformations of the

profile, typically by adding or removing some boxes, vanish at first order. And indeed the

requirement of invariance under adding or removing boxes leads to Bethe-like equations

that characterizes the NS limit [33, 38]. We review this method here.

The instanton partition function of the A2 quiver reads

ZA2 [M1,M2] =∑Y1,Y2

q|Y1|1 q

|Y2|2 Z[M1, Y1]Z[Y1, Y2]Z[Y2,M2], (3.1)

6It was also employed in [41, 42].

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JHEP01(2015)114

where the sum is over all the possible realizations of Ya=1,2 for a fixed set of Coulomb

branch vevs. The objects Ma encode the fundamental flavors content of the theory. They

consist of N(a)f empty Young diagrams, and a vector m

(a)l with l = 1 · · ·N (a)

f corresponding

to the masses of the N(a)f flavors in the fundamental representation. They are such that

R(Ma) = ∅ and A(Ma) = (l, 1, 1), l = 1 · · ·N (a)f . To be able to use the transformation

properties of the previous subsection, we need to require N(1)f = N

(2)f = Nc. In addition,

the bifundamental mass µ is equal to zero for both Z[M1, Y1] and Z[Y2,M2].

The variation of the summand under addition of a box x ∈ A(Y1) can be computed

using the KMZ transformation (2.27) of bifundamental contributions:

q|Y1+x|1 Z[M1, Y1 + x]Z[Y1 + x, Y2]

q|Y1|1 Z[M1, Y1]Z[Y1, Y2]

= − q1

ε1ε2m1(φx)

∏y∈R(Y1)(φx − φy)(φx − φy − ε+)∏y∈A(Y1)y 6=x

(φx − φy)(φx − φy + ε+)

∏y∈A(Y2)(φx − φy − µ+ ε+)∏

y∈R(Y2)(φx − φy − µ).

(3.2)

The mass polynomial ma(x) can be found in (4.2) below, it is monic, with zeros at the value

of the fundamental masses x = m(a)l for l = 1 · · ·N (a)

f . To recover the Bethe equations from

the conditionq|Y1+x|1 Z[M1, Y1 + x]Z[Y1 + x, Y2]

q|Y1|1 Z[M1, Y1]Z[Y1, Y2]

= 1, (3.3)

we assume that the configurations of Young diagrams that contribute in the NS limit are

such that all columns have a different height. As a consequence, a box can be added or

removed from any column. Furthermore, the columns heights are sent to infinity, such that

the products ε2λ(l,a)i remain finite. We then identify the Bethe roots with the instanton

positions of the boxes on top of each column. These also coincide with the position of

boxes that can be added and removed, up to a negligible shift in ε2:

ur = tl,i = φx, x ∈ A(Y ) or R(Y ), in the NS limit. (3.4)

Such an identification was actually already made in [29, 33]. The assumption on the shape

of Young diagrams will be justified by the consistency observed with the results of the Mayer

cluster expansion. There is however a simple heuristic explanation: to keep tl,i finite as

ε2 → 0, we effectively rescaled the Young diagrams columns by 1/ε2 which increases the

disparities between them.7

There is however a small additional subtlety, since there is always one more box that

can be added to a Young diagram than that can be removed. For each Ya, these Nc

extra boxes lie on the right of the diagrams, at positions ξ(a)l = a

(a)l + n

(a)l ε1. In (3.4),

we have specified the set A(Y ) = A(Y ) \ x ∈ Yφx = ξl, excluding the extra boxes at

position ξl. If we are considering Young diagrams with infinitely many columns, implying

that the number of Bethe roots is also infinite, these extra boxes can be neglected and

7Columns with equal heights produce Bethe roots spaced of ε1, which should correspond to strings

solutions of the Bethe equations. A priori, there are no such solutions for the Bethe equations (3.5), but it

could be interesting to perform a deeper analysis of some degenerate situations.

– 10 –

JHEP01(2015)114

A(Y ) ' A(Y ). Here, we will keep the number of columns finite, and n(a)l will act as a

cut-off, as in [30, 31, 43].

To each node of the quiver is associated a set of Bethe roots. For the A2 quiver, we

denote the two sets of Bethe roots ur and vr for the nodes 1 and 2 respectively, with

r = 1 · · ·N (a)B . The number of Bethe roots at each node is N

(a)B = ]A(Ya) = ]R(Ya). From

the identification (3.4), we find for φx → ur that the r.h.s. of (3.2) becomes8

q1m1(ur)ξ1(ur)

N(1)B∏s=1s 6=r

ur − us − ε1ur − us + ε1

N(2)B∏s=1

ur − vs +m+ ε1/2

ur − vs +m− ε1/2= 1,

q2m2(vr)ξ2(vr)

N(2)B∏s=1s 6=r

vr − vs − ε1vr − vs + ε1

N(1)B∏s=1

vr − us −m+ ε1/2

vr − us −m− ε1/2= 1.

(3.5)

The second equation is obtained from the variation of the profile of Y2, it is identical

to the first one with ur and vr exchanged, and m → −m. These two sets of equations

form the so-called Bethe-like equations of the A2 quiver. The factors ξa(x), given by

(m21 = −m12 = −m)

ξa(x) =

Nc∏l=1

∏b 6=a(x− ξ

(b)l +mab + ε1/2)

(x− ξ(a)l )(x+ ε1 − ξ(a)

l ), (3.6)

disappear for infinitely large Young diagrams.

The Young diagrams with infinite profiles are the natural states on which the SHc

generators act in the NS limit. Let us consider a regular object Y (with a finite number

of boxes), and pick a box x = (l, i, j) ∈ A(Y ) ∪ R(Y ) with coordinates (i, j) ∈ Y (l). It is

shown in the appendix B.1 that Λx(Y ) = O(√ε2) unless i = 1 which gives Λx(Y ) = O(1).

Thus, the summation of boxes x ∈ A/R(Y ) defining the action (2.11) of D±1(z) on the

states |Y > can be replaced by a much simpler summation over the number of colors. The

argument breaks down for Young diagrams of infinite profile, for which the difference of

height between two neighboring columns times ε2 becomes macroscopic. Then, Λx(Y ) is of

order O(1/√ε2) for all boxes in A(Y )∪R(Y ), and the action of D±1(z) on the corresponding

states involves a non-trivial summation over the whole set of Bethe roots.

To summarize, we have found that for any test function f , we have for the NS limit of

the object Y : ∑x∈A/R(Y )

f(φx)→NB∑r=1

f(ur),∑

x∈A(Y )

f(φx)→NB∑r=1

f(ur) +

Nc∑l=1

f(ξl). (3.7)

In particular, the function Λ(z)2 becomes:

Λ(z)2 → λ(z)2 =

Nc∏l=1

z − ξl + ε1z − ξl

NB∏r=1

(z − ur)2 − ε21(z − ur)2

. (3.8)

The first factor tends to one if we send the cut-offs nl →∞.

8The factor −ε1ε2 comes from the box y ∈ R(Y1) just below x which is such that φy = φx − ε2.

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JHEP01(2015)114

3.2 NS limit of the algebra

In the NS limit, we rescale the generators Dm,n to define d±n,m = εn/2+m2 D±n,m with

n,m ≥ 0. It implies that the limit of the generating series (2.10) reads

ε1/22 D±1(z)→ d±1(z) =

∑n≥0

(z + ε1)−n−1d±1,n, ε2D0(z)→ d0(z) =∑n≥1

(z + ε1)−nd0,n.

(3.9)

The commutation relations (2.12) become

[d0(z), d±1(w)] = ±ε2d±1(w)− d±1(z)

z − w, [d−1(z), d1(w)] = ε2

e(w)− e(z)z − w

, (3.10)

with en = εn2En and E(z) → e(z). To obtain the relation between the modes en and d0,n

from (2.14) we need to take the limit of the functions πn(s) and ωn(s) with ε2/s fixed.9

We deduce

1 + ε1e(z)=Π(z) exp

−∑n≥1

n(z + ε1)−n [Gn(1+ε1/(z + ε1))+Gn(1− ε1/(z + ε1))] d0,n

,(3.12)

with the function Π(z) depending on the finite rescaled central charges εn2cn:

Π(z) = exp

∑n≥0

(−1)n+1εn2cn(z + ε1)−nGn(1 + ε1/(z + ε1))

=

Nc∏l=1

z + ε1 − alz − al

. (3.13)

The current and Virasoro modes scale as

J±n = (−√−ε1)−nd∓n,0, ε2L±n =

1

n(−√−ε1)−nd∓n,1 +

1

2(1− n)Nε1J±n. (3.14)

The rescaling of the Liouville field by 2b in the semiclassical limit indeed brings a factor

ε2 to the Virasoro modes since Tcl(z) = 4b2Tqu(z) and b ∼ √ε2. We note that most of the

correlators are now vanishing at leading order in ε2, including [d±1(z), d±1(w)] = O(ε2).

This is to be expected since ε2 plays the role of ~ ∼ b2 in the semiclassical limit of Liouville

theory. In this limit, a Virasoro algebra with c = 1 is still present, but the commutators

must be replaced by Poisson brackets [44]. It would be extremely interesting to investigate

this phenomenon more deeply, but it is out of the scope of this paper.

Our choice of rescaling for d±1,n renders the KMZ transformation (2.26) finite,

√−ε1δ±1,zZ[Y1, Y2] = u±1,zZ[Y1, Y2], (3.15)

9We find for η = ε2/s = z + ε1:

πn(s)→ εn2 η−nGn(1 + ε1/η),

ωn(s)→ −(n+ 1)εn+12 η−n−1 [Gn+1(1 + ε1/η) +Gn+1(1− ε1/η)] .

(3.11)

– 12 –

JHEP01(2015)114

where we absorbed the factor√ε2 in the definition of δ±1,z, and the functions defined

in (2.25) simplify into U±1,z → u±1,z with:

u−1,z =

Nc∏l=1

z +m+ ε1/2− ξ(2)l

z − ξ(1)l

∏r

z − ε1 − urz − ur

∏s

z +m+ ε1/2− vsz +m− ε1/2− vs

− 1,

u1,z =

Nc∏l=1

z + ε1 − ξ(1)l

z +m+ ε1/2− ξ(2)l

∏r

z + ε1 − urz − ur

∏s

z +m− ε1/2− vsz +m+ ε1/2− vs

− 1.

(3.16)

The factors containing ξ(a)l cancel each-other in the limit n

(a)l →∞.

The SHc algebra is rather disappointing in the NS limit since all commutators vanish.

It is however possible to obtain non-trivial commutation relations under the rescaling of

d0(z) by ε−12 and d±1(z) by ε

−1/22 :

[d0(z), d±1(w)] = ±d±1(w)− d±1(z)

z − w, [d−1(z), d1(w)] =

e(w)− e(z)z − w

. (3.17)

The commutation relations for [d±1(z), d±1(w)] will also be of order one. Note however

that the relation (3.12) between e(z) and d0,n generators simplifies: the r.h.s. becomes Π(z)

which is independent of the d0,n generators: en are now simple central charges. We expect

this algebra to be realized by some operators of the integrable system. In that respect, it

is satisfying to see that the infinite number of modes d0,n remain commuting, and might

be identified with the conserved charges built upon Dunkl operators in qNLS.

3.3 Covariance of the bifundamental contribution under SHc transformations

The previous description of the NS limit for the objects Ya allows to derive the limit of the

bifundamental contribution Z[Y1, Y2] defined in (2.2). This quantity depends on two sets

of Bethe roots, and will be denoted z[u, v]. The derivation is a bit lengthy, it is done in the

appendix B.2. The main technicality is to get rid of the dependence in the dual variables

tl,j . The final expression is10

z[u, v] =

N(1)B∏r=1

Nc∏l=1

g(ur−ξ(2)l +m+ε1/2)√g(ur−ξ(1)l )g(ur−ξ(1)l +ε1)

N(2)B∏r=1

Nc∏l=1

g(vr−ξ(1)l −m+ε1/2)√g(vr−ξ(2)l )g(vr−ξ(2)l +ε1)

(3.18)

×N

(1)B∏

r,s=1

(g(ur−us−ε1)

g(ur−us+ε1)

)1/4

×N

(2)B∏

r,s=1

(g(vr−vs−ε1)

g(vr−vs+ε1)

)1/4

×N

(1)B∏r=1

N(2)B∏s=1

g(ur−vs+m+ε1/2)

g(ur−vs+m−ε1/2).

This expression involves the function g(x) defined as g(x) = xx/ε2 . This function has a

branch cut on the negative real axis, such that g(e2iπx) = e2iπx/ε2g(x). It implies that

the expression we provided for z[u, v] is ambiguous, and the sheet where x lies in g(x)

must be specified. In the next section, the logarithm of z[u, v] will be related to a Yang-

Yang functional that involves, in addition to the two sets of Bethe roots u(a)r (u

(1)r = ur,

10It still contains a trivial ε2 dependence since ε2 logZ[Y1, Y2] = O(1) as ε2 → 0.

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JHEP01(2015)114

u(2)r = vr), two sets of integers η

(a)r that can be associated to the sheets of the function

g(x). We will further justify that this ambiguity is irrelevant for the KMZ transformation.

The function g(x) satisfies the property g(x ± ε2) ' g(x)x±1e±1 at first order in ε2.

It allows to study the variations of z[ur, vs] under an infinitesimal shift of one Bethe root:

ur → ur ± ε2δr,r′ . The constant factors e±1 cancel each other in the ratios, and we get

z[ur ± ε2δr,r′ , vs]z[ur, vs]

=

Nc∏l=1

(ur′ − ξ(2)l +m+ ε1/2)±1[

(ur′ − ξ(1)l )(ur′ − ξ

(1)l + ε1)

]± 12

N(1)B∏

s′=1s′ 6=r′

(ur′ − us′ − ε1ur′ − us′ + ε1

)± 12

×N

(2)B∏

s′=1

(ur′ − vs′ +m+ ε1/2

ur′ − vs′ +m− ε1/2

)±1

. (3.19)

A similar expression is obtained by considering shifts of the Bethe roots vs, exchanging the

role of ur and vs and flipping the sign of the bifundamental mass.

From now on, we send the cut-offs n(a)l = L → ∞ such that the first product in the

previous expression vanishes. In order to define the SHc generators, we need to introduce

a decomposition of the function λ(z)2 as a sum over its poles, as we did for Λ(z)2 in (2.7):

λ(z)2 = 1 + ε1∑r

∂

∂ur

(λ2r

z − ur

), λ2

r = −ε1∏s 6=r

(ur − us)2 − ε21(ur − us)2

. (3.20)

The factors λr play a role equivalent to Λx(Y ) in the NS limit since Λx(Y ) ' λr/√ε2. The

information of the states |Y > is encoded in the Young diagrams structure. In the NS

limit, this information is rendered by the knowledge of the set of Bethe roots ur. It is

then natural to introduce a state |u > which is an eigenstate of d0(z) with the eigenvalue

d0(z)|u >=

(−

NB∑r=1

log(z − ur) +

Nc∑l=1

nl∑i=1

log(z − al − (i− 1)ε1)

)|u > . (3.21)

This definition of the action of d0(z) will become clear in the next section. The second

term in the r.h.s. acts as a regulator to ensure d0(z) ' ε2|Y |/z at infinity.

The generators constituting e(z) are also diagonal in the basis |u >, with

[1 + ε1e(z)] |u >= λ(z)2|u > (3.22)

inherited from (2.18). Finally, the action of D±1(z) involves the addition or subtraction of

boxes to the Young diagrams. These variations of the columns height generates shifts of

the Bethe roots. Using an exponential notation for shift operators, we define

d±1(z) =∑r

e±ε2∂rλr

z − ur, ∂r =

∂

∂ur. (3.23)

In this definition, we employed the ‘Dunkl’ operator ur such that ur|u >= ur|u >. It satis-

fies with the shift operators the commutation relation [ur, e±ε2∂r ] = ±ε2e±ε2∂r . The opera-

tor λr is obtained from λr by replacing ur with ur, it satisfies [λr, e±ε2∂s ] = ±ε2e±ε2∂s∂sλr.

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JHEP01(2015)114

An explicit calculation shows that the operators d±1(z) and d0(z) obey the commutation

relations (3.10) of SHc at first order in ε2. In addition, expanding over z, one can show

that the generators d0,n and en are related as in (3.12).11

The action of d±1(z) on the bifundamental contributions is obtained from (3.19). In

this expression, we recognize the decomposition of u±1,z over the poles at z = ur. Defining

the KMZ transformation as in (2.23):

δ±1,z = ∓d∓1(z)|ur ± d±1(z +m± ε1/2)|vr , (3.25)

where we indicated on which set of Bethe roots the operator is actually acting, we recover

exactly the KMZ transformation (3.15). It is interesting to note that in order to satisfy

the SHc commutation relations, it is sufficient to expand in ε2 the shift operators in the

definition of d±1(z), and keep only the terms of order O(ε2). However, the full exponential

is needed to obtain the KMZ transformation.

4 Connections with the Mayer cluster expansion

4.1 Integral expression of the A2 partition function

The instanton partition function of N = 2 gauge theories has been originally obtained as

a set of coupled contour integrals [46]. In the case of the A2 quiver,

ZA2 =

∞∑N1,N2=0

qN11 qN2

2

N1!N2!

∫ N1∏i=1

N2∏j=1

K12(φi,1−φj,2)∏a=1,2

Na∏i,j=1i<j

K(φi,a−φj,a)Na∏i=1

Qa(φi,a)dφi,a2iπ

,(4.1)

with qa = ε+qa/ε1ε2. The integration contours over the instanton positions φi,a lie along

the real axis. They are closed in the upper half plane but avoid possible singularities at

infinity. The potential attached to a node a is a ratio involving the mass polynomial ma(x)

and the gauge polynomial Aa(x). These monic polynomials have zeros respectively at the

value of the masses m(a)f of fundamental matter fields, and at the Coulomb branch vevs a

(a)l :

Qa(x) =ma(x)

∏b 6=aAb(x+ ε+/2 +mab)

Aa(x)Aa(x+ ε+),

with ma(x) =

Nc∏f=1

(x−m(a)f ), Aa(x) =

Nc∏l=1

(x− a(a)l ).

(4.2)

As in the first section, we have assumed that both nodes have a gauge group U(Nc) and

Nc fundamental flavors. The kernel K is associated to the nodes, and K12 to the arrow

11The expansion of d0(z) gives

d0,n|u >=1

n

NB∑r=1

[(ur + ε1)n − (xr + ε1)n]|u >, (3.24)

with xr = al + (i − 1)ε1 for r = (l, i) with l = 1 · · ·Nc and i = 1 · · ·nl. This expression can be plugged

into (3.12), and the summation over n performed with the help of (2.17). It reproduces the function λ(z)2.

– 15 –

JHEP01(2015)114

1→ 2. The latter depends on the bifundamental mass m = m12 = −m21. The expressions

of the kernels K and K12 can be found in [28, 47], they are reproduced in the appendix C.

They both depend on the Ω-background equivariant parameters ε1, ε2 which are assumed

purely imaginary. Coulomb branch vevs have a positive imaginary part, and fundamental

masses m(a)f are real.

The integrals can be evaluated as a sum over residues, and poles are known to be in

one to one correspondence with the boxes of the set of objects Ya through the map (2.3).

The resulting expression for the A2 quiver has been given in (3.1). However, the integral

definition reveals more convenient for taking the NS limit. This procedure has been de-

scribed in [28, 29]. We employ here the method of [29], which has been performed only in

the case of a single gauge group. It is easily extended to arbitrary quivers, this is done in

the appendix C.

In the appendix C, two different expressions of the partition function in the NS limit

are provided (eq. (C.14) and (C.19)). The first one will be studied in the next subsection,

it reproduces the familiar form of the Yang-Yang functional [45]. The underlying Bethe

equations are related to those derived in the third section, thus bringing a justification for

the assumption made on the shape of Young diagrams in the NS limit. The bifundamental

contribution z[u, v] given in (3.18) can be recovered from the Yang-Yang functional, and

we will be able to comment on its ambiguities. The second expression for the NS partition

function is derived in the appendix C.2, and will be studied in the subsection 4.3. It consists

of a sum of coupled integrals reminding of a matrix model. The integration variables are

interpreted as the position of hadrons in a one dimensional space. They can be formally

related to the Bethe roots, and we will deduce the eigenvalues (3.21) of the operator d0(z).

4.2 NLIE, Bethe roots and Yang-Yang functional

In [25], it was claimed that the free energy at first order in ε2 can be expressed as an

on-shell effective action. This result is derived for the A2 quiver in the appendix C.1. The

action (C.14) depends on two fields indexed by the node a: the density ρa(x) and the

pseudo-energy εa(x). The equations of motion relate these two fields as

2iπρa(x) = − log(

1− qae−εa(x)), (4.3)

and produce for the field εa(x) a set of NLIE,

εa(x) + logQa(x)−∑b

∫Gab(x− y) log(1− qbe−εb(y))

dy

2iπ= 0, (4.4)

with the kernels G11(x) = G22(x) = G(x) and G21(x) = G12(−x),

G(x) = ∂x log

(x+ ε1x− ε1

), G12(x) = ∂x log

(x+m− ε1/2x+m+ ε1/2

). (4.5)

In the NLIE (4.4), the integration is done over the same contour as in the original defini-

tion (4.18) of the partition function ZA2 .

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JHEP01(2015)114

The set of NLIE (4.4) is characteristic of the Thermodynamical Bethe Ansatz technique

developed in [26, 45]. It is usually associated to the following system of Bethe equations,

q1Q1(ur)

N(1)B∏s=1

ur − us − ε1ur − us + ε1

N(2)B∏s=1

ur − vs +m+ ε1/2

ur − vs +m− ε1/2= 1,

q2Q2(vr)

N(2)B∏s=1

vr − vs − ε1vr − vs + ε1

N(1)B∏s=1

vr − us −m+ ε1/2

vr − us −m− ε1/2= 1.

(4.6)

Indeed, the method elaborated in [27] shows that the counting functions ηa(x) obey the

NLIE (4.4) with an appropriate contour of integration.12 This method is briefly reviewed

in [29]. In particular, it was observed that the derivative of the field ρa(x) corresponds to

a density of Bethe roots:13

− d

dxρa(x) = ρ

(a)B (x), for ρ

(a)B (x) =

N(a)B∑r=1

δ(x− u(a)r ), (4.9)

with the shortcut notations u(1)r = ur and u

(2)s = vs.

Comparing the Bethe equations (4.6) with those obtained in the second section, we

observe that they match only in the formal case Nc = 0. The mismatch comes from missing

factors in the node potential (3.5). However, if we decompose the set of Bethe roots u(a)r

into a set of fixed variables x(a)l,i = a

(a)l + (i − 1)ε1 with i = 1 · · ·n(a)

l and l = 1 · · ·Nc,

and a set of free variables u(a)r , then we observe that the Bethe equations (3.5) for u(a)

r become the Bethe equations (4.6) for u(a)

r . Thus, to each solution ur, vs of the Bethe

equations (4.6) corresponds a solution ur, vs of (3.5) with the additional roots given by

x(1)l,i , x

(2)l′,i′.

14

12Counting functions ηa(x) are defined such that they take integer values at the Bethe roots x = u(a)r ,

for instance

2iπη1(x) = log

q1Q1(x)

N(1)B∏s=1

x− us − ε1x− us + ε1

N(2)B∏s=1

x− vs +m+ ε1/2

x− vs +m− ε1/2

. (4.7)

They are identified with the pseudo-energy as 2iπηa(x) = −εa(x) + log qa.13Since we are dealing with contour integrals, the definition of the delta function may require some

clarification. Here, it is defined as the operator∫δ(x− y)f(y)dy = f(x) (4.8)

for x inside the integration contour (i.e. Im x > 0) which will always be the case.14It follows that N

(a)B = N

(a)B + l(Ya) where l(Ya) =

∑l n

(a)l is the total number of columns in Ya. Thus,

this observation is only valid if we relax the conditions that the number of Bethe roots ur is equal to the

number of columns, otherwise the fixed roots will exhaust all the possible Bethe roots. Here, we use this

observation at the formal level of densities to relate the fields of the NLIE to the Bethe equations. At the

moment, the interpretation of the Bethe equations (4.6) with only a finite set of Bethe roots is not clear in

the gauge theory.

– 17 –

JHEP01(2015)114

It results from this identification that the relation (4.9) among densities is modified

for the Bethe roots u(a)r ,

− d

dxρa(x) = ρ

(a)B (x)− ρ(pert)

a (x),

for ρ(a)B (x) =

N(a)B∑r=1

δ(x− u(a)r ), ρ(pert)

a (x) =

Nc∑l=1

n(a)l∑i=1

δ(x− x(a)l,i ).

(4.10)

Inserted in the integrals of (4.4) through the relation (4.3), this amended identity gives an

expression of the pseudo-energy in terms of the Bethe roots u(a)r where Qa(x) is replaced

with ma(x)ξa(x).15 It allows to identify 2iπηa(x) = −εa(x) + log qa with the counting

function associated to the Bethe equations (3.5). Those equations are obtained by imposing

ηa(u(a)r ) ∈ Z, i.e. e2iπηa(u

(a)r ) = 1. Interestingly, the correction term in (4.10) reproduces the

part of the density involved in the perturbative contributions to the gauge theory partition

function [43]. It allows to interpret the Bethe equations (3.5) of the second section as the

extremum of the full N = 2 partition function, including the perturbative contribution.16

On the other hand, the Bethe equations (4.6) obtained here describe the extremization of

the instanton contribution only.

We now examine the free energy FA2 = limε2→0 ε2 logZA2 derived from the on-shell

action (C.14). Instead of the pseudo-energies εa(x), we prefer to work with the counting

functions ηa(x),

FA2 =∑a=1,2

Fa[ρa] + F12[ρ1, ρ2] +∑a=1,2

J[ρa, ηa]. (4.12)

In the r.h.s., we have singled out the parts that depend on ηa:

J[ρ, η] = −2iπ

∫ρ(x)η(x)dx+

1

2iπ

∫Li2

(e2iπη(x)

)dx, (4.13)

and decomposed the remaining term into three contributions:

Fa[ρa] =1

4

∫ρa(x)ρa(y)G(x−y)dxdy+

∫ρa(x) log

(qama(x)√

Aa(x)Aa(x+ε1)

)dx,

F12[ρ1, ρ2] =1

4

∑a=1,2

∫ρa(x)ρa(y)G(x−y)dxdy+

∫ρ1(x)ρ2(y)G12(x−y)dxdy

+∑a=1,2

∫ρa(x) log

(Ab6=a(x+mab + ε1/2)√

Aa(x)Aa(x+ ε1)

)dx.

(4.14)

In these expressions, the densities ρa(x) can be replaced by the Bethe roots density ρ(a)B (x)

using an integration by parts and the relation (4.10), leading to summations over Bethe

15We use the property (1− e−ε1∂x

)ρ(pert)a (x) =

Nc∑l=1

[δ(x− a(a)

l )− δ(x− ξ(a)l )]. (4.11)

16We refer here to the case of Young diagrams with infinitely many columns, such that the factor ξa(x)

is not present.

– 18 –

JHEP01(2015)114

roots ur and vs. In particular, the exponential of F12[ρ1, ρ2] reproduces the expression

found in (3.18) for z[u, v],

e1ε2F12[ρ1,ρ2] ≡ z[u, v], (4.15)

up to ambiguities of the same nature as in section three. Similarly, Fa[ρa] reproduces

the NS limit of the matter bifundamental contributions Z[Ma, Ya]. It is noted that, if

there are several solutions to the Bethe equations, the partition function is a sum over the

contributions of each solution.

We have thus shown that F12 and Fa reproduce the partition function of the A2

quiver derived in the second section. But we have here an additional term J[ρ, η] that

remains to be treated. To do so, we repeat the trick used in [29]: in (4.13), the second

integral simplifies using an integration by parts and the relation between the density and

the counting function derived from (4.3). One of the terms obtained cancel with the first

term in the expression of J[ρ, η], and it only remains

J[ρ, η] = 2iπ

∫xρ′(x)η(x)dx. (4.16)

For the first node, it gives from (4.10) in terms of the Bethe roots:

J[ρ1, η1] = −2iπ∑r

urη(1)r + 2iπ

Nc∑l=1

n(1)l∑i=1

x(1)l,i η1(x

(1)l,i ). (4.17)

with η(1)r = η1(ur) ∈ Z. The second term in the r.h.s. is independent of the Bethe roots,

and simply produces an overall constant factor. The first term is more interesting since it

has the same form as the ambiguity arising in (3.18) from 2iπ rotations of the argument of

the functions g(x). Since η(1)r is an integer, a shift of ur → ur± ε2 leads to a variation δJ =

∓2iπε2η(1)r which vanishes when exponentiated with a factor 1/ε2. Thus, this ambiguity

does not interfere with the KMZ transformation we have defined.

The full expression FA2 [u(a)r , η

(a)r ] reproduces the Yang-Yang functional of qNLS, up to

the potential term. Thus, (3.21) and (3.23) define an action of the SHc operators at first

order in ε2 on the Yang-Yang functional of an integrable system.

4.3 Hadronic integrals

Another expression of the partition function ZA2 in the NS limit has been obtained in the

appendix C.2,

ZA2 '∞∑

p,q=1

ε−p−q2

p!q!

∞∑k1,···kp=1

∞∑l1,··· ,lq=1

q∑i ki

1 q∑j lj

2∏i k

2i

∏j l

2j

∫ p∏i=1

dxi2iπ

q∏j=1

dyj2iπ

I(k, x|l, y), (4.18)

with the integrand

I(k, x|l, y) =∏i

Q1(xi)ki∏j

Q2(yj)lj exp

ε2∑i,j

kiljG12(xi − yj)

× exp

ε22

∑i,j

kikjG(xi − xj) +ε22

∑i,j

liljG(yi − yj)

.

(4.19)

– 19 –

JHEP01(2015)114

This expression is a consequence of a phenomenon of clustering or confinement of the

instantons as ε2 → 0. Indeed, the integrations in the original expressions (4.1) are done

over the positions of N1 + N2 instantons denoted φi,a. It is common in the study of

random matrix models to see the integration variables φi,a as pseudo-particles of color

a, in an external potential logQa, and interacting through the kernels K and K12. In

the NS limit, a number ki of instantons at positions φα,1 (α = 1 · · · ki) may become very

close to each other due to the infinitely strong short-range interaction of K(x). These

instantons, at a distance ∼ ε2 of each-other, can be approximated by a single particle of

charge ki and center of mass xi. By analogy with quarks bound states, it will be called a

hadron. This phenomenon is described with more details in [29]. The factors 1/k2i in the

summation (4.18) are a consequence of the interaction between the elementary instantons

composing a hadron. For the second node, hadrons positions are denoted yj and they have

the charge lj .

The equality (4.18) holds at leading order in ε2. Both sides contain ε2 corrections that

do not match. To extract the leading order in ε2, (generalized) matrix model techniques

can be employed [48]. A Mayer expansion can also be considered, the first order free

energy being expressed as a sum over connected clusters with a tree structure, and vertices

associated to hadrons. The free energy in the NS limit is equal to the Legendre transform

of the canonical free energy at large Na, where Na is the conjugate variable of the chemical

potential log qa. In this setting, only configurations (k, x) with ε2∑

i ki = O(1) contribute

to the leading order. The charges ki are supposed to be finite, and the sum is over i ∈ Z>0.

Thus, we will consider that for any test function f(x), the following summations are of

order one:

ε2

∞∑i=1

kif(xi). (4.20)

4.3.1 Hadronic variables and Bethe roots

In the hadronic formulation, the pseudo-energy is seen as a Lagrange multiplier enforcing

the following identities,

ρ1(x) = ε2

p∑i=1

kiδ(x− xi), ρ2(y) = ε2

q∑j=1

ljδ(y − yj), (4.21)

where the field ρa(x) is defined as the dressed vertex of the Mayer expansion. Although

pertaining to different contexts, it is instructive to formally identify these densities with the

fields of the previous subsection, and use (4.10) to provide a bridge between the expressions

in terms of (ki, xi) and ur variables.

It is possible to give another argument in favor of the relation (4.10) between variables

(ki, xi) and ur. Let f(x) be a test function without singularities in the upper half plane.17

The summation (4.20) of kif(xi) can be interpreted as an operator evaluated in the statis-

tical average that defines the NS partition function (4.18). As such, it is the NS limit of

17Typically f(x) is a polynomial in x, or the generating function 1/(z − x) for Im z < 0.

– 20 –

JHEP01(2015)114

the operatorε1ε2ε+

∑i

f(φi,1) (4.22)

evaluated as the vev associated to the original partition function (4.1). This vev can also

be computed as a sum over residues, and since f(x) has no singularities within the contour

of integration, we findε1ε2ε+

∑x∈Y1

f(φx), (4.23)

with the object Y1 defined in the first section, and the map x→ φx in (2.3). Thus, in the

NS limit, we can formally replace

ε1ε2ε+

∑x∈Y1

f(φx)→ ε2∑i

kif(xi), (4.24)

and similarly for Y2 and the variables (lj , yj). The consistency with the identification of

Bethe roots as the position of boxes on Young diagrams edges (3.7) implies18

NB∑r=1

f(ur) = ε2∑i

kif′(xi) +

Nc∑l=1

nl∑i=1

f(xl,i). (4.27)

This is the equivalent of the relation (4.10) for the densities.

There is a simple heuristic interpretation of the relation we have observed between the

variables (ki, xi) and ur. It boils down to represent the Young diagrams columns as made of

an infinite number of hadrons. For simplicity, let us assume Nc = 1 with a unique Coulomb

branch vev a1 = a, and NB = n1 = n. The generalization to Nc > 1 is straightforward.

We drop the color and node indices, and consider the expression

NB∑r=1

[f(ur)− f(xr)] =n∑r=1

[f(xr + λrε2)− f(xr)], (4.28)

due to the identification of ur with t1,r, and xr = x1,r = a + (r − 1)ε1. The difference of

the two functions in the r.h.s. can be replaced by an integral of the derivative,

NB∑r=1

[f(ur)− f(xr)] =

n∑r=1

∫ λr

0f ′(xr + ε2tr)ε2dtr. (4.29)

18As an example, we can use the identity (2.19) that provides an alternative expression of the function

Λ(z)2 as a product over the boxes in Y . As ε2 → 0, the function r(z) defined in (2.9) is expended as

r(z) = 1 + ε2θ(z) +O(ε22), θ(z) = ∂z log

(z2

z2 − ε21

). (4.25)

Performing the NS limit of the l.h.s. of (2.19) as in (4.24), and expressing the r.h.s. with (3.8) in terms of

Bethe roots, we get

N(1)B∏r=1

(z − ur)2 − ε21(z − ur)2

=

Nc∏l=1

(z + ε1 − al)(z − ξl)(z − al)(z + ε1 − ξl)

eε2∑

i kiθ(z−xi), (4.26)

in agreement with (4.27). Note that this identity has a well defined limit when ξl →∞.

– 21 –

JHEP01(2015)114

... ...

Figure 2. Decomposition of an infinite column (here rotated by 90) into finite sets of kr,s boxes.

In the NS limit, each column λr contains an infinite number of boxes. We divide this infinite

amount of boxes into infinitely many finite sets of size kr,s, ranging from ur,s =∑

s′<s kr,s′

to ur,s + kr,s:

NB∑r=1

[f(ur)− f(xr)] =n∑r=1

∞∑s=1

∫ ur,s+kr,s

ur,s

f ′(xr + ε2tr,s)ε2dtr,s

= ε2

n∑r=1

∞∑s=1

∫ kr,s

0f ′(xr,s + ε2tr,s)dtr,s,

(4.30)

with xr,s = xr + ε2ur,s, ur,0 = 0 and ur,∞ = λr. Notations are displayed on figure 2. By

the mean value theorem, for each interval there exists 0 ≤ kr,s ≤ kr,s such that

∫ kr,s

0f ′(xr,s + ε2tr,s)dtr,s = kr,sf

′(xr,s), xr,s = xr,s + ε2kr,s. (4.31)

The mean value xr,s depends on the function f . However, for functions of order O(1), the

variation of xr,s with f is of order O(ε2) and thus can be neglected. It would not be the

case if the kr,s were infinite, and indeed it is not possible to assign an average instanton

position to a hadron with infinite charge because of its macroscopic size. Plugging (4.31)

into (4.30), we recover the relation (4.27) for any test function f(x) provided we identify

(kr,s, xr,s) ≡ (ki, xi). Thus, the relation between Bethe roots and hadronic variables

corresponds to the decomposition of the Young diagrams columns into finite sets of ki boxes

with an average instanton position xi.

4.3.2 Relation between bifundamental summands and integrands

The formal relation we have observed between the variables (ki, xi) and the Bethe roots

enables the comparison between the integral expression (4.18) and the Bethe roots sum-

mation of the A2 quiver partition function. The decomposition (3.1) of the summands can

be reproduced on the integrand I(k, x|l, y), with three parts that will be the analogue of

the three bifundamental contributions Z[M1, Y1], Z[Y1, Y2] and Z[Y2,M2] respectively:

I(k, x|l, y) = z1(k, x)z12(k, x|l, y)z2(l, y) with:

z1(k, x) =∏i

m1(xi)ki

(A1(xi)A1(xi + ε1))ki/2exp

ε24

∑i,j

kikjG11(xi − xj)

,

– 22 –

JHEP01(2015)114

z12(k, x|l, y) =∏i

A2(xi +m+ ε1/2)ki

(A1(xi)A1(xi + ε1))ki/2

∏j

A1(yj −m+ ε1/2)lj

(A2(yj)A2(yj + ε1))lj/2

× exp

ε24

∑i,j

kikjG11(xi−xj)+ε2∑i,j

kiljG12(xi−yj)+ε24

∑i,j

liljG22(yi−yj)

,z2(l, y) =

∏j

m2(yj)lj

(A2(yj)A2(yj + ε1))lj/2exp

ε24

∑i,j

liljG22(yi − yj)

. (4.32)

The factors z1(k, x) and z2(l, y) depend only on one set of variables and can be attached to

the nodes a = 1, 2. On the other hand, z12(k, x|l, y) is related to the arrow 1→ 2 and will

be referred as the bifundamental integrand. Up to a pure mass term, the factor z1(k, x)

can be obtained as z12(k, x|l′, y′) for a specific choice of variables (l′, y′) such that

ε2∑j

l′jG12(z − y′j) =∑f

log(z −m(1)f ). (4.33)

These variables (l′, y′) encode the information of the object M1, in the same way that (k, x)

encodes Y1. Similar ideas can also be found in [43] in the language of densities.

It is remarkable that performing the change of representation (k, x)→ ur (and (l, y)→vs), materialized by the relation (4.10) among densities, the bifundamental integrand

z12(k, x|l, y) becomes z[u, v], the NS limit of the bifundamental contribution (3.18). This

formal identification may allow to define the KMZ transformation on the integrand expres-

sions. In particular, the infinitesimal shifts of the Bethe roots ur → ur ± ε2 correspond

to a variation of the density ρB(x) → ρB(x) ∓ ε2δ′(x − ur) which is rendered by a shift

of ki → ki ± 1 for the hadron of coordinate xi ' ur + O(ε2). The main difficulty lies

in the decomposition of the function λ(z)2 over poles, since in the representation (k, x)

this function is a product of essential singularities. It is still possible to use the Cauchy

identity to write it as a sum over contributions at the location of the singularities, but the

evaluation of the integrals is problematic.

On the other hand, the definition of d0(z), diagonal on the states |k, x >= |u > easily

follows from the definition (2.10) of D0(z), and the limit (4.24), which gives, taking into

account the ε2 factor discrepancy in (3.9) between d0 and D0:

d0(z)|k, x >= ε2∑i

kiz − xi

|k, x > . (4.34)

This expression can be integrated and expressed in terms of Bethe roots using (4.10). It

leads to the formula (3.21) used in the previous section.

5 Discussion

In this paper, we considered the NS limit of the KMZ transformation that represents the

action of the SHc generators on the bifundamental contribution to the A2 quiver instanton

partition function. In this limit, the bifundamental contribution can be written in terms

– 23 –

JHEP01(2015)114

of two sets of Bethe roots. We have defined a set of generators of degree 0,±1 that

exploits an infinitesimal variation of the Bethe roots. These generators reproduce the

proper commutation relations, and generates an equivalent of the KMZ transformation. We

then turned to an alternative approach that uses the Mayer cluster expansion to perform

the NS limit. The coordinates and charges of the hadronic variables that appear in this

formalism were formally related to the Bethe roots. This provides a hint on the possibility

to realize the KMZ transformation directly on integral expressions. Finally, we studied a

third expression of the partition function that takes the form of a Yang-Yang functional.

Although obtained via Mayer expansion, this expression coincides with the limit of the

Young diagrams summations under a proper identification of the Bethe roots. We deduced

an action of the SHc generators on the functional (at first order in ε2).

Now that the limit of the algebra has been identified, it remains to relate it to the

algebraic structures of the underlying integrable systems, either the Yangian of XXX spin

chains or the DDAHA of qNLS. The connection between SHc and the Calogero-Moser

Hamiltonian for Nc = 1 allows to identify the states |Y > with Jack polynomials. It

is thus natural to expect a connection between the states |u > and the eigenfunctions

of the integrable systems. Such a connection could be obtained using the fact that the

Calogero-Moser Cherednik operators reduce to the Dunkl operators of qNLS in a specific

limit [20]. This limiting process may be the equivalent of the NS limit from the integrable

point of view.

A better understanding of the states |u > may also arise from the study of the single

gauge group case (A1 quiver). The connections between SHc and Gaiotto states [49] has

been discussed recently in [50]. Some of these considerations may survive in the NS limit,

and help to identify a semi-classical version of the Gaiotto states. We hope to address

these issues in a near future.

Acknowledgments

I would like to thank Davide Fioravanti for valuable comments at the early stage of this

work. I am also indebted to Yutaka Matsuo for several fruitful discussions and many ad-

vices. It is a pleasure to acknowledge the Tokyo University (Hongo) for the kind hospitality

and support during my stay there. I also want to thank Sergio Andraus Robayo for a very

interesting discussion on Dunkl and intertwining operators. I acknowledge the Korea Min-

istry of Education, Science and Technology (MEST) for the support of the Young Scientist

Training Program at the Asia Pacific Center for Theoretical Physics (APCTP).

A Computing with instanton positions

A.1 Transition of notations

To derive the dictionary between the notations of [2] and ours, let us drop for a moment

the color and node indices, and focus on a single Young diagram. In [2], Young diagrams

were encoded as a sequence of f rectangles characterized by the integers 0 < r1 < · · · < rfand 0 < sf < · · · < s1. It is further assumed that r0 = sf+1 = 0. The transition between

– 24 –

JHEP01(2015)114

notations is done as follows: to each box x ∈ A(Y ) with x = (l, i, j), corresponds an index

k ∈ [[1, f + 1]] such that i = rk−1 + 1 and j = sk + 1 (the labels l and a are implicit on

f, rk, sk, · · · ). Defining Ak(Y ) = βrk−1 − sk − ξKMZ as in [2], we deduce

al,KMZ +Ak(Y ) = −(φx + ε+)/ε2, (A.1)

where al,KMZ denotes the Coulomb branch vevs in [2], rescaled here as al,KMZ = −al/ε2. In

a similar way, for x ∈ R(Y ), there is a k ∈ [[1, f ]] such that i = rk and j = sk, and defining

Bk(Y ) = βrk − sk we find

al,KMZ +Bk(Y ) = −(φx + ε+)/ε2, (A.2)

with the same rescaling of the Coulomb branch vevs.

A.2 Useful formulas

In some computations, it is not necessary to resort to the rectangle decomposition of Young

diagrams. Here we provide some useful formulas in this respect. Considering the difference

between sets of boxes that can be added or removed, we find

∑x∈A(Y )

φx −∑

x∈R(Y )

φx =

Nc∑l=1

(al + ε+f(l)),

∑x∈A(Y )

φx(φx − ε+)−∑

x∈R(Y )

φx(φx + ε+) =∑l

al(al − ε+)− 2ε1ε2|Y |,(A.3)

where f (l) is the number of rectangles in the Young diagram Y (l) and |Y | the total number

of boxes.

Let us also mention the identities coming from the expansion of (2.7) at z →∞:∑x∈A(Y )

Λx(Y )2 −∑

x∈R(Y )

Λx(Y )2 = Nc,

∑x∈A(Y )

φxΛx(Y )2 −∑

x∈R(Y )

φxΛx(Y )2 =∑l

al +1

2ε+Nc(Nc − 1),

∑x∈A(Y )

φ2xΛx(Y )2 −

∑x∈R(Y )

φ2xΛx(Y )2 = −2ε1ε2|Y |+

∑l

a2l

+ ε+(Nc−1)∑l

al+1

6ε2+Nc(Nc−1)(Nc−2).

(A.4)

The action of the commutators of two degree (minus) one operators on states |Y >

may be used to define some of the higher generators:

[D±1(z), D±1(w)]|Y >=∑

x∈A/R(Y )

∑y∈A/R(Y±x)

Λx(Y )Λy(Y ±x)(z−w)(φx−φy)(z−φx)(z−φy)(w−φx)(w−φy)

|Y ±x±y > .

(A.5)

– 25 –

JHEP01(2015)114

B Justifications

B.1 Order of Λx(Y )

Let x ∈ A(Y ) such that x = (l, i, j) with (i, j) ∈ Y (l). The multiplicity of a column λ(l)i of

Y (l) is the number of columns in Y (l) with the same height λ(l)i . There is no other y ∈ A(Y )

in the same column, i.e. such that φy = φx+O(ε2). But there is always a unique y ∈ R(Y )

such that φy = φx − ε+ + O(ε2) (except for i = 1). There is a unique y1 ∈ R(Y ) with

φy1 = φx + O(ε2), and a unique y2 ∈ A(Y ) with φy2 = φx + ε+ + O(ε2), if and only if

the column λ(l)i is of multiplicity one. Thus, Λx(Y ) = O(

√ε2) except if i = 1 for which

Λx(Y ) = O(1).

Now, we let x = (l, i, j) ∈ R(Y ). There is no other y ∈ R(Y ) with φy = φx + O(ε2),

but there is always a unique y ∈ A(Y ) such that φy = φx + ε+ +O(ε2). There is a unique

y ∈ A(Y ) such that φy = φx+O(ε2) if and only if λ(l)i is of multiplicity one. Likewise, there

is a unique y ∈ R(Y ) such that φy = φx − ε+ + O(ε2) if and only if λ(l)i is of multiplicity

one, and i > 1. Thus, if i 6= 1, we have Λx(Y ) = O(√ε2). If i = 1, λ

(l)1 is of multiplicity

one, and Λx(Y ) = O(1).

B.2 NS limit of the bifundamental contribution

In this appendix, we take the NS limit of the bifundamental contribution (2.2). A similar

calculation can be found in [51]. The cut-offs n(a)l are assumed to be all equal to L which

will be sent to infinity at the end of the computation. We first concentrate on a single

object Y , and examine the following quantity that is involved in the denominator:

P (x) =∏

(l,i,j)∈Y

Nc∏l′=1

(tl,j − tl′,i + x). (B.1)

In the NS limit, tl′,i is identified with the Bethe root ur, and the main difficulty is to treat

the dual quantities tl,j . For this purpose, let us focus on the lth Young diagram pictured

in figure 3, and consider the first column. Boxes in the set denoted (a) are such that

tl,j = al + ε1 + (j − 1)ε2 for j = λ(l)2 + 1 · · ·λ(l)

1 . Similarly, boxes in the set (b) are mapped

to tl,j = al + 2ε1 + (j− 1)ε2 with j = λ(l)3 + 1 · · ·λ(l)

2 . In general, the first column should be

divided into nl = L sets of boxes with tl,j = al + rε1 + (j − 1)ε2, j = λ(l)r+1 + 1 · · ·λ(l)

r with

r = 1 · · ·L and λ(l)L+1 = 0. The same applies to the next columns, but we should start at

r = i for the ith column. We thus have found

P (x) =

Nc∏l,l′=1

L∏i=1

L∏r=i

λ(l)r∏

j=λ(l)r+1+1

(tl,j − tl′,i + x). (B.2)

Plugging in the explicit expression for tl,j , we obtain

P (x) =

Nc∏l,l′=1

L∏i=1

L∏r=i

ελ

(l)r −λ

(l)r+1

2

λ(l)r∏

j=λ(l)r+1+1

(yll′ir + (j − 1)), ε2yll′ir = al + rε1 + x− tl,i. (B.3)

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JHEP01(2015)114

...

Figure 3. Young diagrams with a labeling of boxes in the first column (we dropped the node and

color indices).

The product over j can now be replaced by a ratio of gamma functions,

P (x) =

Nc∏l,l′=1

L∏i=1

L∏r=i

ελ

(l)r −λ

(l)r+1

2

Γ[yll′ir + λ(l)r ]

Γ[yll′ir + λ(l)r+1]

. (B.4)

As ε2 → 0, the gamma functions arguments tend to infinity, and we can use the Stirling

approximation Γ[x] ' (x/e)x (the square root is subleading). Noticing that ε2(yll′ir+λ(l)r ) =

tl,r − tl′,i + x + ε1 and ε2(yll′ir + λ(l)r ) = tl,r+1 − tl′,i + x, and introducing the function

g(x) = xx/ε2 , we can write

P (x) 'Nc∏l,l′

L∏r,i=1r≥i

eλ(l)r+1−λ

(l)rg(tl,r − tl′,i + x+ ε1)

g(tl,r+1 − tl′,i + x). (B.5)

This expression now involves only the variables tl,i which can be replaced by Bethe roots.

It is however better to slightly shift the products indices to get

P (x) ' e−Nc|Y |Nc∏l,l′

L∏i=1

g(tl,i − tl′,i + x+ ε1)

g(tl,L+1 − tl′,i + x)×

Nc∏l,l′=1

L∏r,i=1r>i

g(tl,r − tl′,i + x+ ε1)

g(tl,r − tl′,i + x). (B.6)

The previous result upon P (x) can be used to take the NS limit of the denominator

factors in (2.2). Considering

Q2 =∏

(l,i,j)∈Y

Nc∏l′=1

(tl,j − tl′,i + ε2)(tl,j − tl′,i − ε1), (B.7)

we find

Q2 ' e−2Nc|Y |Nc∏l,l′

L∏i=1

g(tl,i−tl′,i)g(tl,i−tl′,i+ε1)

g(tl,L+1−tl′,i)g(tl,L+1−tl′,i−ε1)×

Nc∏l,l′=1

L∏r,i=1r>i

g(tl,r−tl′,i+ε1)

g(tl,r−tl′,i−ε1). (B.8)

– 27 –

JHEP01(2015)114

It is possible to use the property g(e±iπx)g(x) = e±iπx/ε2 to symmetrize the product. An

ambiguity arises in the choice of the sense of the rotation. It is a part of the overall

ill-definiteness for the final result of z[u, v]. Here, we make the choices that provide the

simplest expressions: we rotate both numerator and denominators in the ratios in the same

direction, e+iπ. After re-arranging the diagonal terms r = i, we obtain

Q2 ' e−2Nc|Y |eiπN2cL

2ε1/2ε2

Nc∏l,l′=1

L∏i=1

g(tl′,i − tl,L+1)g(tl′,i − tl,L+1 + ε1)

×Nc∏l,l′=1

L∏r,i=1

(g(tl,r − tl′,i + ε1)

g(tl,r − tl′,i − ε1)

)1/2

.

(B.9)

Finally, replacing tl,i → ur and tl,L+1 → ξl (NB = NcL), we find:

Q2'e−2Nc|Y |eiπN2Bε1/2ε2

Nc∏l=1

NB∏r=1

g(ur−ξl)g(ur−ξl+ε1)×NB∏r,s=1

(g(ur−us+ε1)

g(ur−us−ε1)

)1/2

. (B.10)

A similar treatment can be applied to the numerator, but we now have to keep track of

the node indices. This does not add much difficulty, and we find for the numerator of (2.2)

the expression

e−Nc|Y1|−Nc|Y2|eiπN2Bε1/2ε2

Nc∏l=1

N(1)B∏r=1

g(ur − ξ(2)l +m+ ε1/2)

×Nc∏l=1

N(2)B∏r=1

g(vr − ξ(1)l −m+ ε1/2)×

N(1)B∏r=1

N(2)B∏s=1

g(ur − vs +m+ ε1/2)

g(ur − vs +m− ε1/2).

(B.11)

Taking the ratio, we end up with the formula given in (3.18).

C Nekrasov-Shatashvili limit from Mayer expansion

C.1 Equations of motion, grand-canonical free energy

In this appendix, we study a general class of grand-canonical partition functions defined

as the discrete Laplace transform

ZGC(q1, · · · , qM ) =∞∑

N1,··· ,NM=0

M∏a=1

(qa/ε)Na

Na!ZC(N1, · · ·NM ) (C.1)

of the following canonical partition functions

ZC(N1, · · ·NM )=

∫ M∏a,b=1a<b

Na∏i=1

Nb∏j=1

Kab(φi,a−φj,b)M∏a=1

Na∏i,j=1i<j

Kaa(φi,a−φj,a)Na∏i=1

Qa(φi,a)dφi,a2iπ

.(C.2)

– 28 –

JHEP01(2015)114

Partition functions with M = 1 were considered in [29], and we show here that the method

extends smoothly to M > 1. The integration contour is along the real axis, closed in the

upper half plane, and avoiding possible singularities at infinity. We study the limit ε→ 0,

with the kernels defined as

Kab(x) = 1 + εGab(x) + εδabpa(x), pa(x) =αaε

x2 − ε2, (C.3)

and Gab(x) a function independent of ε. Note that the poles at x = ±ε that pinch the

integration contour are only present in the diagonal term Kaa, the non-diagonal kernel Kab

with a 6= b is not singular at ε→ 0. This is in agreement with quivers instanton partition

functions where Kab corresponds to the bifundamental contribution (with a non-zero mass

mab), and Kaa to the vector hypermultiplet.19 We will further assume that Gab(x) satisfies

the property Gab(x) = Gba(−x). This condition is necessary to be able to expand over

non-oriented clusters.20 In particular, this implies that the diagonal entries Gaa(x) are

even functions of x.

It is possible to express the integrand of the partition function (C.2) with the help

of double indices I = (i, a) (and J = (j, b)) for a = 1 · · ·M and i = 1 · · ·Na, and with

the lexicographical order I < J if and only if a < b or (a = b and i < j). Applying the

Mayer expansion technique [52–54] to (C.1) for Kab = 1 + εfab, we find for the free energy

FGC = ε logZGC:

FGC =

∞∑l1,···lM=0

∑Cl

ε−(l−1)

σ(Cl)

∫ ∏I∈V (Cl)

qaQa(φI)dφI2iπ

∏<IJ>∈E(Cl)

εfab(φI − φJ). (C.4)

The connected clusters Cl are made of l =∑

a la vertices I = (i, a). We denote by V (Cl)

the set of vertices of the cluster Cl, and E(Cl) the set of links < IJ > connecting two

vertices I and J . Each vertex bear a color label a, and there are la vertices of type a.21

The symmetry factor σ(Cl) of the cluster is the cardinal of the group of automorphisms

preserving the cluster. These automorphisms must also preserve the color labeling because

the combinatorial factor in the definition of ZGC is∏aNa! and not (

∑aNa)! as in the usual

Mayer cluster expansion. Decomposing fab = Gab + δabpa, the cluster expansion becomes

a sum over clusters Cl with two types of links: G-links with kernel Gab between vertices of

color a and b, and p-links of kernel pa between two vertices of the same color a:

FGC =

∞∑l1,···lM=0

∑Cl

ε−(l−1)

σ(Cl)

∫ ∏I∈V (Cl)

qaQa(φI)dφI2iπ

∏<IJ>∈Ep(Cl)

αaε2

(φI − φJ)2 − ε2

×∏

<IJ>∈EG(Cl)

εGab(φI − φJ).

(C.5)

19For the application to the A2 quiver partition function, we take Kaa(x) = ∆(x)∆(−x) and Kab(x) =

1/∆(x + mab − ε+/2) with ∆(x) = x(x + ε+)/((x + ε1)(x + ε2)) [47]. It gives in the NS limit with ε = ε2,

Kaa(x) = 1 + εG(x) + εpa(x) +O(ε2) (with αa = 1) and K12(x) = 1 + εG12(x) +O(ε2). The functions G(x)

and G12(x) are defined in (4.5).20It corresponds to the condition Sab(x)Sba(−x) = 1 for the scattering amplitudes, Gab(x) =

−∂x logSab(x), and can be obtained as a consequence of unitarity and hermitian analyticity.21Let us emphasize that the color of the vertices corresponds to the node index a of the quiver. It has

nothing to do with the gauge color index of fields in the gauge theory.

– 29 –

JHEP01(2015)114

We have E(Cl) = Ep(Cl) ∪ EG(Cl) where Ep denotes the set of p-links, and EG the set of

G-links. The clusters that contribute at first order in ε have no cycle involving G-links,

but do have cycles of only p-links. They are of order O(εl−1) so that FGC defined above is

of order one [29].

The generating function of rooted vertices, also called dressed vertex, bears an index

a corresponding to the color of the root,

Y (a)(x) = qaQa(x)

∞∑l1,···lM=0

∑Cx,al

ε−(l−1)

σ(Cx,al )

∫ ∏I∈V (C

x,al

)

I 6=x

qaQa(φI)dφI2iπ

×∏

<IJ>∈Ep(Cx,al )

εpa(φIJ)∏

<IJ>∈EG(Cx,al )

εGab(φIJ),

(C.6)

where we denoted φIJ = φI−φJ . The summation is over rooted clusters Cx,al , with the root

x = φx of color a. The symmetry factor σ(Cx,al ) ≤ σ(Cl) is the number of automorphisms

of Cl that leave the root invariant. The generating functions Y(a)G (x) and qaQa(x)Y

(a)p (x)

are defined with the additional requirement that the root x is linked to other vertices with

only G- or p- links respectively. At first order, we have the factorization property

Y (a)(x) ' Y (a)G (x)Y (a)

p (x), (C.7)

as a consequence of the property that G-links do not form cycles.

Minimal clusters that contribute to Y(a)G can be decomposed into sub-clusters rooted

by the direct descendants. The root x, of type a, is linked to mb vertices φj,b of type b

through a link Gab, which leads to

Y(a)G (x) = qaQa(x)

∞∑m1,··· ,mM=0

M∏b=1

ε−mb

mb!

mb∏i=1

∫εGab(x− φi,b)Y (b)(φi,b)

dφi,b2iπ

. (C.8)

The symmetry factors mb! take into account the possibility of permuting vertices of the

same color. Contributions of sub-clusters factorize to give

Y(a)G (x) = qaQa(x) exp

(M∑b=1

∫Gab(x− y)Y (b)(y)

dy

2iπ

). (C.9)

This is the generalization of the formula (2.6) of [29].

The third relation is obtained by repeating the confinement argument employed in [29].

This argument remains unchanged because p-links only relate vertices of the same color.

It gives the following relation between Y (a)(x) and Y(a)G (x):

Y (a)(x) = lαa(Y(a)G (x)), (C.10)

with the function lα(x) defined in equ (2.13) of [29]. For α = 1, this function simplifies

into a logarithm: l1(x) = − log(1− x).

– 30 –

JHEP01(2015)114

The fields ρa and ϕa are introduced as

Y (a)(x) = 2iπρa(x), Y(a)G (x) = qaQa(x)e−ϕa(x) ⇒ 2iπρa(x) = lαa(qaQa(x)e−ϕa(x)).

(C.11)

Exploiting the relations (C.9) and (C.10), we obtain the TBA-like NLIE:

ϕa(x) +∑b

∫Gab(x− y)lαb(qbQb(y)e−ϕb(y))

dy

2iπ= 0. (C.12)

For αa = 1, we recover the NLIE (4.4) with the pseudo-energies εa defined as εa(x) =

ϕa(x)− logQa(x).

Back to the free energy. The grand-canonical (instanton) free energy is obtained from

the technique displayed in [29] simply by introducing an additional sum over the different

colors. The expression of FGC follows from the Basso-Sever-Vieira formula, FGC = Γ0 −(1/2)Γ1 with

Γ0 =M∑a=1

∫dx

2iπLαa(Y

(a)G (x)), Γ1 =

M∑a,b=1

∫Y (a)(x)Y (b)(y)Gab(x− y)

dxdy

(2iπ)2. (C.13)

The function Lα(x) is a primitive of lα(x), it reduces at α = 1 to the dilogarithm L1(x) =

Li2(x). Plugging in the equations of motion, and using the variable ρa and ϕa instead of

Y (a) and Y(a)G , we recover the results of [28]: FGC = SGC[ρa, ϕa] on-shell with the effective

action

SGC[ρa, ϕa] =1

2

M∑a,b=1

∫ρa(x)ρb(y)Gab(x− y)dxdy +

M∑a=1

∫ρa(x)ϕa(x)dx

+1

2iπ

M∑a=1

∫Lαa(qaQa(x)e−ϕa(x))dx. (C.14)

The equations of motion reproduce the relations (C.11) and (C.12).

C.2 Confinement

For simplicity, in this section we focus on the case M = 2, but the generalization to higher

M is straightforward. To obtain the hadronic partition function, we mimic the treatment

performed in [29]. At first order in ε2, the instanton partition function can be written as a

path integral,

ZGC(q1, q2) '∫D[ρa, ϕa]e

1εSGC[ρa,ϕa]. (C.15)

The function Lαa contains all the dependence in qa, expanding it we get

exp

(1

ε

∫Lαa

(qaQa(x)e−ϕa(x))dx

2iπ

)=

∞∑p=1

ε−p

p!

∞∑k1,k2,··· ,kp=1

∫ p∏i=1

I(a)kiqkia Qa(xi)

kie−kiϕa(xi)dxi2iπ

,

(C.16)

– 31 –

JHEP01(2015)114

with I(a)k the coefficients of the Taylor expansion of Lαa(x):

Lαa(x) =∞∑k=1

I(a)k xk. (C.17)

Plugging this expansion in the expression of ZGC(q1, q2), we find

ZGC(q1, q2) '∞∑

p,q=1

ε−p−q

p!q!

∑kipi=1

∑ljqj=1

∫ p∏i=1

I(1)kiqki1 Q1(xi)

kidxi2iπ

q∏j=1

I(2)ljqlj2 Q2(yj)

ljdyj2iπ∫

D[ρa, ϕa]e12ε

∑a,b

∫ρa(x)ρb(y)Gab(x−y)dxdye

1ε

∫ϕ1(x)(ρ1(x)−ε

∑i kiδ(x−xi))dx

× e1ε

∫ϕ2(y)(ρ2(y)−ε

∑j ljδ(y−yj))dy. (C.18)

The fields ϕa appear as Lagrange multipliers enforcing the conditions (4.21) for the densities

(with ε = ε2). Replacing the densities, we find

ZGC(q1, q2) '∞∑

p,q=1

ε−p−q

p!q!

∑kipi=1

∑ljqj=1

∫ p∏i=1

I(1)kiqki1 Q1(xi)

kidxi2iπ

q∏j=1

I(2)ljqlj2 Q2(yj)

ljdyj2iπ

(C.19)

× exp

ε

2

∑i,j

kikjG11(xi − xj) +ε

2

∑i,j

liljG11(yi − yj) + ε∑i,j

kiljG12(xi − yj)

.

The expression given in (4.18) is obtained after specialization of (C.1) to the quiver par-

tition function (4.1). It implies to set ε = ε2, α1 = α2 = 1 leading to I(a)k = 1/k2. The

kernels G11 and G22 coincide with the function G given in (4.5).

Open Access. This article is distributed under the terms of the Creative Commons

Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in

any medium, provided the original author(s) and source are credited.

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